A082985 → basis A013610

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0:    1;
  1:    4,      9;
  2:   11,     45,      45;
  3:   29,    168,     315,     189;
  4:   76,    567,    1539,    1782,     729;
  5:  199,   1815,    6435,   10989,    8910,     2673;
  6:  521,   5616,   24570,   55458,   67392,    41067,     9477;
  7: 1364,  16965,   88290,  247860,  402165,   372519,   178605,   32805;
  8: 3571,  50337,  303705, 1019898, 2072385,  2586006,  1908522,  743580,  111537;
  9: 9349, 147288, 1010781, 3951639, 9652608, 15176079, 15208398, 9224766, 2991816, 373977;

The row polynomials satisfy P(n,x) = -(1+6*x+9*x^2)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 9*x.

P[0, x_] := 1;
P[1, x_] := 4 + 9*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 9*x^2)*P[n - 2, x] + (3 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 6*x + 9*x^2)*v[i - 2] + (3 + 6*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-d584905ee0ee13fd
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-d584905ee0ee13fd
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-d584905ee0ee13fd
• Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  The roots of consecutive polynomials alternate on the real line.
  by the cited steps: e-d584905ee0ee13fd

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• basis_map A013610 e-d584905ee0ee13fd

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check three_term_recurrence e-4fa62233b528a616 root: theorem check

      Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
2. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023